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User blog:B1mb0w/J Function Sandpit J 1
'J Function Sandpit \(J_1\)' The J Function is a work in progress. This sandpit defines a function called \(J_1\) which contains ideas that will be used in the final J Function. Click here for the J Function blog. 'Definition' \(J_1\) uses an algorithm to generate \(J_0\) functions and ascending ordinal values. A simple illustration of the algorithm is the following expression: \(J_1(2^n) = J_0(n,n^n) = f_{\epsilon_0}(n)\) The next expression defines the general case better. A real number r is converted into two input parameters n (an integer) and s (a real number) for the \(J_0\) function. \(J_1® = J_0(n,s)\) where n is an integer and r and s can be a real numbers Example code for the \(J_1\) function is provided at the end of this blog. Examples and definitions of the the \(J_0\) function are available here 'Calculated Examples up to \(J_1®\)' \(J_1(0) = f_{\omega}(1)\) \(J_1(1) = f_{\omega}(1)\) \(J_1(2) = f_{1}(2)\) \(J_1(3) = f_{\omega + 1}(2)\) \(J_1(4) = f_{\omega + 2}(3)\) \(J_1(5) = f_{\omega^{\omega}.2}^{2}(3)\) \(J_1(6) = f_{\omega^{\omega.2 + 1}.2 + 1}(3)\) \(J_1(7) = f_{\omega^{\omega^{2} + \omega + 2}.2 + 2}^{2}(3)\) \(J_1(7.5) = f_{\omega^{\omega^{2}.2 + \omega}}(3)\) \(J_1(7.75) = f_{\omega^{\omega^{2}.2 + \omega.2}.2 + \omega.2 + 2}^{2}(3)\) \(J_1(7.99) = f_{\omega^{\omega^{\omega}}}(3)\) \(J_1(8) = f_{\omega^{\omega.3} + 3}(4)\) \(J_1(9) = f_{\omega^{\omega^{3}.2 + \omega^{2}.2}}(4)\) \(J_1(10) = f_{\omega^{\omega^{\omega + 3}.3 + \omega^{2}}}(4)\) \(J_1(11) = f_{\omega^{\omega^{\omega.3 + 1} + 2} + \omega.3}(4)\) \(J_1(12) = f_{\omega^{\omega^{\omega^{2} + \omega.2 + 2}.2 + \omega.2}}(4)\) \(J_1(13) = f_{\omega^{\omega^{\omega^{2}.2 + \omega.3 + 3}.2 + \omega^{2}.2}}(4)\) \(J_1(14) = f_{\omega^{\omega^{\omega^{3} + \omega^{2} + \omega + 1}}}(4)\) \(J_1(15) = f_{\omega^{\omega^{\omega^{3}.2 + \omega^{2}.2 + \omega.2 + 1}}}(4)\) \(J_1(16) = f_{\omega^{\omega^{\omega^{2} + 4}.4 + 1}}(5)\) \(J_1(17) = f_{\omega^{\omega^{\omega^{3}.2 + \omega^{2}.4 + \omega.2 + 4}.4 + \omega^{3}.4 + \omega^{2}.4 + \omega.4 + 4}.4 + \omega^{3}.4 + \omega^{2}.4 + \omega.4 + 4}^{4}(5)\) \(J_1(18) = f_{\omega^{\omega^{\omega^{4}.4 + \omega^{3}.3 + \omega^{2} + \omega.4 + 4}.4 + \omega^{2}.4 + \omega.4 + 4}.4 + \omega^{3}.4 + \omega^{2}.4 + \omega.4 + 4}^{4}(5)\) \(J_1(20) = f_{\omega^{\omega^{\omega^{\omega.3 + 1} + 2}.3 + \omega.4 + 1}.4 + \omega^{3}.4 + \omega^{2}.4 + \omega.4 + 4}^{4}(5)\) WORK IN PROGRESS 'VBA Program Code' The following is VBA visual basic code and will run as a macro in Microsoft Excel. This function creates a string literal of a \(J_0\) Function. Example results are listed above. Function j_1(r As Double) As String Dim i As Integer, d As Double i = jLog(2, r) + 1 d = i - 1 d = d ^ d d = jReal * (i ^ i - d) + d d = Int(d + 0.5) j_1 = i & " - " & d j_1 = jString(i, d) End Function How the Function Works A description of how the code works will be provided here ... Work in Progress. *VBA Constants *VBA Data Structures *VBA Functions **'j_1' Function - This function extracts takes the Log Base 2 from a real number. The integer part is used to set the \(n\) parameter of the \(J_0\) function. The decimal part is then re-calibrated against a range up to the value of \(n^n\) and this becomes the s parameter of the \(J_0\) function. **'jLog' Function - is defined here **'jString' Function - is defined here *''Work In Progress'' Category:Blog posts